<abstract xmlns="http://www.w3.org/1999/xhtml">

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.
</abstract>

LAGA, Université Sorbonne Paris Nord et CNRS

Aix Marseille Univ, CNRS, Centrale Marseille

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we prove that if <italic>a</italic>, <italic>b</italic> > 0 and 0 ≤ <italic>v</italic> ≤ 1. Then for all positive integer <italic>m</italic>
(1) - For <italic>v</italic> ∈ <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>v</mi><mo>∈</mo><mrow><mo>[</mo> <mrow><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup></mrow></mfrac></mrow> <mo>]</mo></mrow></mrow></math>
<tex-math>v \in \left[ {0,{1 \over {{2^n}}}} \right]</tex-math>
</alternatives>
</inline-formula>, we have
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>a</mi></mrow><mi>v</mi></msup><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>v</mi></mrow><mi>m</mi></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><msup><mrow><mi>b</mi></mrow><mi>m</mi></msup></mrow></msqrt><mo>-</mo><mroot><mrow><mrow><mo>(</mo><mrow><mi>a</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>m</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mi>k</mi></msup></mrow></mroot></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>≤</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>v</mi><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>.</mo></mrow></mrow></math>
<tex-math>{\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.}</tex-math>
</alternatives>
</disp-formula>
(2) - For <italic>v</italic> ∈ <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>v</mi><mo>∈</mo><mrow><mo>[</mo> <mrow><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup></mrow></mfrac><mo>,</mo><mn>1</mn></mrow> <mo>]</mo></mrow></mrow></math>
<tex-math>v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right]</tex-math>
</alternatives>
</inline-formula>, we have
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_004.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>a</mi></mrow><mi>v</mi></msup><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><msup><mrow><mi>a</mi></mrow><mi>m</mi></msup></mrow></msqrt><mo>-</mo><mroot><mrow><mrow><mo>(</mo><mrow><mi>b</mi><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>m</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mi>k</mi></msup></mrow></mroot></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>≤</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>v</mi><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>v</mi></mrow><mo>)</mo></mrow><mi>b</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>m</mi></msup><mo>,</mo></mrow></mrow></math>
<tex-math>{\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},}</tex-math>
</alternatives>
</disp-formula>
we also prove two similar inequalities for the cases <italic>v</italic> ∈ <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_005.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>v</mi><mo>∈</mo><mrow><mo>[</mo> <mrow><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup></mrow></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow> <mo>]</mo></mrow></mrow></math>
<tex-math>v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right]</tex-math>
</alternatives>
</inline-formula> and <italic>v</italic> ∈ <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0015_eq_006.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>v</mi><mo>∈</mo><mrow><mo>[</mo> <mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup><mo>+</mo><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mi>n</mi></msup></mrow></mfrac></mrow> <mo>]</mo></mrow></mrow></math>
<tex-math>v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right]</tex-math>
</alternatives>
</inline-formula>. These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and <italic>p</italic>-norms of positive τ-measurable operators.
</abstract>

Further generalized refinement of Young’s inequalities for τ -mesurable operators

<abstract xmlns="http://www.w3.org/1999/xhtml">

This paper deals with a new class of mixed fractional differential equations with integral boundary conditions. We show an important equivalence result between our problem and nonlinear integral Fredholm equation of the second kind. The existence and uniqueness of a positive solution are proved using Guo-Krasnoselskii’s fixed point theorem and Banach’s contraction mapping principle. Different types of Ulam-Hyers stability are discussed. Three examples are also given to show the applicability of our results.
</abstract>

A new class of mixed fractional differential equations with integral boundary conditions

<abstract xmlns="http://www.w3.org/1999/xhtml">

The main goal of this paper is to provide the maximum number of crossing limit cycles of two different families of discontinuous piecewise linear differential systems. More precisely we prove that the systems formed by two regions, where, in one region we define a linear center and in the second region we define a Hamiltonian system without equilibria can exhibit three crossing limit cycles having two or four intersection points with the cubic of separation. After we prove that the systems formed by three regions, where, in two noadjacent regions we define a Hamiltonian system without equilibria, and in the third region we define a center, can exhibit six crossing limit cycles having four and two simultaneously intersection points with the cubic of separation.
</abstract>

Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics

<abstract xmlns="http://www.w3.org/1999/xhtml">

We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.
</abstract>

Department of Mathematics, Faculty of Sciences El Jadida

Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz

Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and <italic>L</italic>1-data

<abstract xmlns="http://www.w3.org/1999/xhtml">

This paper aims to propose a new hybrid approach for solving multiobjective optimization problems. This approach is based on a combination of global and local search procedures.
The cross-entropy method is used as a stochastic model-based method to solve the multiobjective optimization problem and reach a first elite set of global solutions. In the local search step, an ∈-constraint method converts the multiobjective optimization problem to a series of parameterized single-objective optimization problems. Then, sequential quadratic programming (SQP) is used to solve the derived single-objective optimization problems allowing to reinforce and improve the global results. Numerical examples are used to demonstrate the efficiency and effectiveness of the proposed approach.
</abstract>

Combining the cross-entropy algorithm and ∈-constraint method for multiobjective optimization

<abstract xmlns="http://www.w3.org/1999/xhtml">

The purpose of the present paper is to find the sufficient conditions for the subclasses of analytic functions associated with Pascal distribution to be in subclasses of spiral-like univalent functions and inclusion relations for such subclasses in the open unit disk 𝔻. Further, we consider the properties of integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.
</abstract>

Certain subclasses of Spiral-like univalent functions related with Pascal distribution series

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.
</abstract>

Engineering Sciences and Energy Management Laboratory

Laboratory of Engineering Sciences, Faculty of Science

Technical Research Laboratory, Faculty of Applied Sciences

A Partition of unity finite element method for valuation American option under Black-Scholes model

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we solve and investigate the generalized additive functional inequalities
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0022_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow></mrow></math>
<tex-math>\left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|</tex-math>
</alternatives>
</disp-formula>
and
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0022_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>.</mo></mrow></math>
<tex-math>\left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|.</tex-math>
</alternatives>
</disp-formula>
Using the direct method, we prove the Hyers-Ulam stability of the functional inequalities (0.1) in Banach spaces and (0.2) in non-Archimedian Banach spaces.
</abstract>

Department of Mathematics, Faculty of Sciences Applied

Department of Mathematics and Computer Science

Department of Mathematics, Faculty of Sciences

Generalized functional inequalities in Banach spaces

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Moroccan Journal of Pure and Applied Analysis

 The Moroccan Journal of Pure and Applied Analysis "MJPAA" (Print ISSN: 2605-6364) is a peer-reviewed open access journal published by the "Université Sidi Mohamed Ben Abdellah" (Sidi Mohamed Ben Abdallah University), covering all main-stream branches of pure and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, numerical analysis, optimization theory, and control theory.   MJPAA is published by Sidi Mohamed Ben Abdallah University in three issues in January, May and September.   Why subscribe and read    The journal covers a wide range of branches of mathematical analysis.   Pure and applied analysis topics are intensely studied by researchers throughout the world especially in Africa, the Arab and European countries.   An edition committee of high-level and international renown.   Editorial board representing different regions of the world and various research teams.      Why submit    No publication charges for the authors or their institutions.   The journal is available to be indexed in the most important databases.   The journal is among the best in its field in Africa and the Arab world.      Rejection Rate    45%      Archiving   Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.   Plagiarism Policy   The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.   Publisher:Sidi Mohamed Ben Abdallah University (Route Imouzzer, BP 2626 Fez 30000, Morocco, Tel. : 0212 5 35 60 96 60/ 61)   Contact:Professor Youssef Akdim, Sidi Mohamed Ben Abdellah Universitymjpaa.edition1@usmba.ac.ma 